Corrigendum for
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S. Katsumata, E. Rivas, T. Uustalu. Interaction laws of monads and
comonads.
Proc. of LICS '20, pp. 604-618. ACM Press, 2020.
doi:10.1145/3373718.3394808
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p 5, col 1, first parag of Sec 2.4:
The parenthetical remark "(e.g., to finitary functors if \C is locally
finitely presentable)" is false, is in the text by mistake. It holds
about the coends for Day convolution in Sec 5.1, but not about the
ends here.
The dual of G exists and is accessible for an accessible G if \C is
locally presentable. Hence restricting to the full subcategory of
[\C,\C] given by accessible endofunctors is a good choice in this
case.
The sentence "For (-)^\circ to be a contravariant...", which repeats
what is stated in the previous sentences, is in the text by mistake.
p 5, col 2 and p 6, col 1, whole of Sec 2.5:
For most constructions here, Cartesian closedness of \C is not enough,
well-pointedness is needed (or one should define interaction laws
as strong natural transformations between strong endofunctors).
Specifically, well-pointedness is needed for the following:
- Dual of the identity functor
- Dual of terminal functor
- Duals of exponentials of the identity functor
- Example 2.4
- Duals of exponentials of a general functor
- Dual of composition of general functors
- Example 2.5
Well-pointedness is not needed for:
- Duals of products of a functor, initial functor, coproduct of two
functors.
There is a tension of whether interaction laws should be defined as
(ordinary) natural transformations or strong natural
transformations. There are advantages to interaction laws as strong
(ie enriched) natural transformations instead of ordinary natural
transformations, but also disadvantages. For a discussion, see Sec 6
of McDermott, Rivas, Uustalu, FoSSaCS 2022.